Must I have cemeteries / landfills or can I use crematoriums / incinerators exclusively?

Yes, you need both cemeteries and crematoriums, as some citizens will refuse to be cremated. (Source : Cities Skylines guide)

For trash this is not necessary, you can suffice with just incinerators as trash doesn't really have an opinion on how it is disposed off.

For both it is important to build enough facilities spread out trough the city as they have a limited number of trucks/hearses and these need to drive trough traffic to the pick up location.